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The present paper is concerned with the existence of response solutions of quasi-periodic type for a class of quasi-periodically forced, non-Hamiltonian but reversible nonlinear beam equations. We do not suppose the basic frequency ω ∈ R 2 of the forcing term is Diophantine or Brjuno, and it might be Liouvillean, which is weaker than the Diophantine or Brjuno frequency. The proof is based on an improved Kolmogorov–Arnold–Moser (KAM) theorem for infinite-dimensional reversible systems with non-reducible normal form. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of the initial-nonlinear nonlocal solutions for a parabolic system in a weighted Sobolev space. The methods applied are the Faedo–Galerkin approximation and the general theory of weak compactness in appropriate weighted Sobolev spaces together with using the Poincaré-type operator for dealing nonlinear nonlocal conditions. Furthermore, the boundedness and positivity of solutions depending on the boundedness and positivity of given data are also discussed by using suitable test functions. [ABSTRACT FROM AUTHOR]

This paper concerns the weak unique continuation property of solutions of a general system of differential equation/inequality with a second order strongly elliptic system as its leading part. We put not only some natural assumptions which we call basic assumptions, but also some technical assumptions which we call further assumptions. It is shown as usual by first applying the Holmgren transform to this equation/inequality and then establishing a Carleman estimate for the leading part of the transformed inequality. The Carleman estimate is given via a partition of unity and the Carleman estimate for the operator with constant coefficients obtained by freezing the coefficients of the transformed leading part at a point. A little more details about this are as follows. Factorize this operator with constant coefficients into two first order differential operators. Conjugate each factor by a Carleman weight, and derive an estimate which is uniform with respect to the point at which we froze the coefficients for each conjugated factor by constructing a parametrix for its adjoint operator. [ABSTRACT FROM AUTHOR]

This paper considers the problem of the local existence for the generalized MHD equations with fractional dissipative terms Λ 2 α u for the velocity field and Λ 2 β b for the magnetic field, respectively. Based on some new commutator estimates, local existence for the generalized MHD equations is established, which recovers and improves previous results. [ABSTRACT FROM AUTHOR]

In this paper, we study a non-local coupled system that arises in the theory of dislocations densities dynamics. Based on a new gradient entropy estimate in L log L space, we prove the global existence of a continuous solution. The approach is made by adding a viscosity term and passing to the limit for vanishing viscosity. A comparison principle with respect to time is used for proving uniqueness of the solution for the local problem. [ABSTRACT FROM AUTHOR]

In this paper, we study a non-local coupled system arising in the modeling of the dynamics of dislocation densities in crystals. For this system, the global existence and uniqueness are available only for continuous viscosity solutions. In the present paper, we investigate the global time existence of this system by considering BV initial data. Based on a fundamental uniform BV estimate and the finite speed of propagation property of this system, we show, in a particular setting, the global existence of discontinuous viscosity solutions of this problem. [ABSTRACT FROM AUTHOR]

This paper is devoted to study of a nonlinear heat equation with a viscoelastic term associated with Robin conditions. At first, by the Faedo–Galerkin and the compactness method, we prove existence, uniqueness, and regularity of a weak solution. Next, we prove that any weak solution with negative initial energy will blow up in finite time. Finally, by the construction of a suitable Lyapunov functional, we give a sufficient condition to guarantee the global existence and exponential decay of weak solutions. [ABSTRACT FROM PUBLISHER]

Mechanical degradation of electrode materials, in the form of particle cracking and fragmentation, disintegration, fracturing, and loss in contact between current collectors and the active electrode materials, can affect or deteriorate the performance of Li-ion batteries dramatically and even lead to the battery failure in electric vehicle. This paper firstly built a single particle model (SPM) based upon kinetics of electrochemical reactions. Then the Li-ion concentration, evolution of diffusion induced stresses within the SPM under potentiostatic or galvanostatic operating conditions were analyzed by utilizing a mathematical method. Next, evolution of stresses or strains in the SPM, at the core of relates with mechanical degradation of electrode materials, are elaborated in detail. Finally, surface and morphology of the electrodes dismantled from fresh and degraded cells after galvanostatic charge/discharge cycling have been analyzed to verify the hypothesis aforementioned by observing scanning electron microscopy and analyzing X-ray diffraction. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Robin problem for a wave equation with nonlinear source containing nonlocal term. Using the Faedo-Galerkin method and the linearization method for nonlinear term, the existence and uniqueness of a weak solution are proved. An asymptotic expansion of high order in a small parameter of a weak solution is also discussed. [ABSTRACT FROM AUTHOR]

In this paper we study the initial value problem for the nonlinear wave equation with damping and source terms with some $\rho(x)$ and $f(u)$ on the whole space $\mathbb{R}^{n}$ ( $n\geq 3$ ). For the low initial energy case, which is the non-positive initial energy, based on a concavity argument we prove the blow-up result. As for the high initial energy case, we give sufficient conditions of the initial data such that the corresponding solution blows up in finite time. In other words, our results imply a complete blow-up theorem in the sense of the initial energy, $-\infty< E(0)<+\infty$ . [ABSTRACT FROM AUTHOR]

In this paper, we consider the Robin-Dirichlet problem for a nonlinear wave equation with the source term containing a nonlinear integral. Using the Faedo-Galerkin method and the linearization method for nonlinear terms, we prove the existence and uniqueness of a weak solution. We also discuss an asymptotic expansion of high order in a small parameter of a weak solution. [ABSTRACT FROM AUTHOR]

In this paper, we consider the cauchy problem for the 2D Groma-Balogh model (Acta Mater 47:3647-3654, 1999). From the works Cannone et al. (Arch Ration Mech Anal 196:71-96, 2010) and El Hajj (Ann Inst Henri Poincaré Anal Nonlinéaire 27:21-35, 2010), one can see global well-posedness for this model is an open question. However, we can prove longtime well-posedness. In particular, we show that this model admits a unique solution with the lifespan $$T^\star $$ satisfying $$T^\star \log ^2(1+T^\star )\gtrsim \epsilon ^{-2}$$ if the initial data is of size $$\epsilon $$ . To achieve this, we first establish some new decay estimates concerning the operator $$e^{-{\mathcal {R}}_{12}^2t}$$ . Then, we prove the longtime well-posedness by utilizing the weak dissipation to deal with the nonlinear terms. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we consider the following nonlinear wave equation where , , , are given functions and , are given constants. In Eq. , the nonlinear term depends on the integrals , . In this paper we associate with Eq. a recurrent sequence defined by , , with satisfying . The first term is chosen as . If and , we prove that the sequence converges at a rate of order to a unique weak solution of problem (1). [Copyright &y& Elsevier]

This paper deals with a fully implicit time discretization scheme with variable time-step for a nonlinear system modelling phase transition and mechanical deformations in shape memory alloys. The model is studied in the non-stationary case and accounts for local microscopic interactions between the phases introducing the gradients of the phase parameters. The resulting initial-boundary value problem has already been studied by the author who proved existence, uniqueness and continuous dependence on data for a suitable weak solution along some regularity results. A careful and detailed investigation of the variable time-step discretization is the goal of this paper. Thus, we deduce some estimates for the discretization error. These estimates depend only on data, impose no constraints between consecutive time-steps and show an optimal order of convergence. Finally, we prove another regularity result for the solution under stronger regularity assumptions on data. [ABSTRACT FROM AUTHOR]

This paper is concerned with the asymptotic behavior of global C solutions of the Goursat problem for quasilinear hyperbolic systems. Based on the existence result on the global classical solution, we prove that when t tends to the infinity, the solution approaches a combination of Lipschitz continuous and piecewise C traveling wave solutions, provided that the C norm of the boundary data is bounded but possibly large, and the BV norm of the boundary data is sufficiently small. Applications include the 1D compressible Euler equations for Chaplygin gases. [ABSTRACT FROM AUTHOR]

Abstract: This paper deals with the asymptotic behavior of strong solutions to the 3D Navier–Stokes equations with a nonlinear damping term . First, we establish an upper bound for the difference between the solution of our equation and the heat equation in space. Then, we optimize the upper bound of decay for the solutions and obtain their algebraic lower bound by using Fourier Splitting method. [Copyright &y& Elsevier]

Abstract: In this paper we prove local well-posedness in and for the generalized sixth-order Boussinesq equation . Our proof relies in the oscillatory integrals estimates introduced by Kenig et al. (1991) . We also show that, under suitable conditions, a global solution for the initial value problem exists. In addition, we derive the sufficient conditions for the blow-up of the solution to the problem. [Copyright &y& Elsevier]

Large-scale simulations of coupled flow in deformable porous media require iterative methods for solving the systems of linear algebraic equations. Construction of efficient iterative methods is particularly challenging in problems with large jumps in material properties, which is often the case in realistic geological applications, such as basin evolution at regional scales. The success of iterative methods for such problems depends strongly on finding effective preconditioners with good parallel scaling properties, which is the topic of the present paper. We present a parallel preconditioner for Biot's equations of coupled elasticity and fluid flow in porous media. The preconditioner is based on an approximation of the exact inverse of the two-by-two block system arising from a finite element discretisation. The approximation relies on a highly scalable approximation of the global Schur complement of the coefficient matrix, combined with generally available state-of-the-art multilevel preconditioners for the individual blocks. This preconditioner is shown to be robust on problems with highly heterogeneous material parameters. We investigate the weak and strong parallel scaling of this preconditioner on up to 512 processors and demonstrate its ability on a realistic basin-scale problem in poroelasticity with over eight million tetrahedral elements. [ABSTRACT FROM AUTHOR]

This paper establishes the global existence of classical solution to the system of homogeneous, isotropic hyperelasticity with time-independent external force, provided that the nonlinear term obeys a type of null condition. The authors first prove the existence and uniqueness of the stationary solution. Then they show that the solution to the dynamical system converges to the stationary solution as time goes to infinity. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following nonlinear Kirchhoff wave equation where $\widetilde{u}_{0}$ , $\widetilde{u}_{1}$ , μ, f, g are given functions and $\Vert u_{x}\Vert ^{2}=\int_{0}^{1}u_{x}^{2}(x,t)dx.$ To the problem (1), we associate a linear recursive scheme for which the existence of a local and unique weak solution is proved by applying the Faedo-Galerkin method and the weak compact method. In particular, motivated by the asymptotic expansion of a weak solution in only one, two or three small parameters in the researches before now, an asymptotic expansion of a weak solution in many small parameters appeared on both sides of (1) is studied. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we consider the mixed initial-boundary value problem for quasilinear hyperbolic systems with nonlinear boundary conditions in the first quadrant . Under the assumptions that the system is strictly hyperbolic and linearly degenerate or weakly linearly degenerate, the global existence and uniqueness of solutions are obtained for small initial and boundary data. We also present two applications for physical models. [Copyright &y& Elsevier]

Abstract: We present a new a priori estimate for discrete coagulation–fragmentation systems with size-dependent diffusion within a bounded, regular domain confined by homogeneous Neumann boundary conditions. Following from a duality argument, this a priori estimate provides a global bound on the mass density and was previously used, for instance, in the context of reaction–diffusion equations. In this paper we demonstrate two lines of applications for such an estimate: On the one hand, it enables to simplify parts of the known existence theory and allows to show existence of solutions for generalised models involving collision-induced, quadratic fragmentation terms for which the previous existence theory seems difficult to apply. On the other hand and most prominently, it proves mass conservation (and thus the absence of gelation) for almost all the coagulation coefficients for which mass conservation is known to hold true in the space homogeneous case. [Copyright &y& Elsevier]

Abstract: This paper is concerned with a parabolic equation with a non-local term defined on a compact two-dimensional Riemannian surface . If the total mass of the solution, , is equal to and is the standard sphere , it is a Hamilton’s normalized Ricci flow. We obtain the global in time existence of the solution to this problem for . If , the orbit is compact while for , there is a time sequence along which the solution converges to a stationary solution. [Copyright &y& Elsevier]

Abstract: In this paper, we study the model of Groma and Balogh [I. Groma, P. Balogh, Investigation of dislocation pattern formation in a two-dimensional self-consistent field approximation, Acta Mater. 47 (1999) 3647–3654] describing the dynamics of dislocation densities. This is a two-dimensional model where the dislocation densities satisfy a system of two transport equations. The velocity vector field is the shear stress in the material solving the equations of elasticity. This shear stress can be related to Riesz transforms of the dislocation densities. Basing on some commutator estimates type, we show that this model has a unique local-in-time solution corresponding to any initial datum in the space for and , where is the Hölder–Zygmund space. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate the mixed initial-boundary value problem for diagonalizable quasilinear hyperbolic systems with nonlinear boundary conditions on a half-unbounded domain . Under the assumptions that system is strictly hyperbolic and linearly degenerate, we obtain the global existence and uniqueness of solutions with the bounded norm of the initial data as well as their derivatives and appropriate boundary condition. Based on the existence results of global classical solutions, we also prove that when tends to infinity, the solutions approach a combination of travelling wave solutions. Under the appropriate assumptions of initial and boundary data, the results can be applied to the equation of time-like extremal surface in Minkowski space . [Copyright &y& Elsevier]

Abstract: In this paper, we consider the following nonlinear wave equation where , , , , are given functions. To problem (1), we associate a linear recursive scheme for which the existence of a local and unique weak solution is proved by applying the Faedo–Galerkin method and the weak compact method. In the case of , , , , for all , and , , , a weak solution having an asymptotic expansion of order in two small parameters , is established for the following equation associated to (1) 2,3: [Copyright &y& Elsevier]

In this paper, we show that there are almost periodic solutions corresponding to full dimensional invariant tori for higher dimensional Schrödinger equations with Fourier multiplier i u t −Δ u+ M ξ u+ f(| u|2) u = 0, subject to periodic boundary conditions, where the nonlinearity f is a real-analytic function near u = 0 with f(0) = 0. The proof is based on an improved infinite dimensional KAM theorem. [ABSTRACT FROM AUTHOR]

Abstract: We consider non-isothermal phase separation models of the Penrose–Fife type, which were proposed in [O. Penrose, P.C. Fife, Thermodynamically consistent models of phase-field type for the kinetics of phase transitions, Physica D 43 (1990) 44–62]. In the present paper, we impose a non-homogeneous Dirichlet boundary condition on the nonlinear heat flux. Moreover, we consider two cases as boundary conditions for the conserved order parameter. One is the homogeneous Neumann boundary condition and the other is the Signorini boundary condition, which is nonlinear. We show the existence and uniqueness of solutions to these models. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate the mixed initial–boundary value problem with small BV data for linearly degenerate quasilinear hyperbolic systems with general nonlinear boundary conditions in the half space . As the result in [A. Bressan, Contractive metrics for nonlinear hyperbolic systems, Indiana Univ. Math. J. 37 (1988) 409–421] suggests that one may achieve global smoothness even if the norm of the initial data is large, we prove that, if the norm of the initial and boundary data is bounded but possibly large, and the BV norm of the initial and boundary data is sufficiently small, then the solution remains globally in time. Applications to quasilinear hyperbolic systems arising in physics and mechanics, particularly to the system describing the motion of the relativistic string in the Minkowski space–time , are also given. [Copyright &y& Elsevier]

Abstract: In this paper, the multi-layer quasi-geostrophic ocean model with delays is studied. First, a few criteria are derived for asymptotic stability of the stationary solutions. Then, to quantify the impact of delay on the overall system evolution, an estimate is derived for comparing the quasi-geostrophic ocean states with or without delays. [Copyright &y& Elsevier]

Abstract: The paper studies the longtime behavior of solutions to the initial boundary value problem (IBVP) for a nonlinear wave equation arising in an elastic waveguide model . It proves that when the space dimension , under rather mild conditions the dynamical system associated with the above-mentioned IBVP possesses a global attractor which is connected and has finite fractal and Hausdorff dimension. [Copyright &y& Elsevier]

Abstract: The resemblance between the Horn–Thompson theorem and a recent theorem by Dacorogna–Marcellini–Tanteri indicates that Schur-convexity and the majorization relation are relevant for applications in the calculus of variations and its related notions of convexity, such as rank one convexity or quasiconvexity. In Theorem 6.6, we give simple necessary and sufficient conditions for an isotropic objective function to be rank one convex on the set of matrices with positive determinant. Majorization is used in order to give a very short proof of a theorem of Thompson and Freede [R.C. Thompson, L.J. Freede, Eigenvalues of sums of Hermitian matrices III, J. Res. Nat. Bur. Standards B 75B (1971) 115–120], Ball [J.M. Ball, Constitutive inequalities and existence theorems in nonlinear elastostatics, in: R.J. Knops (Ed.), Nonlinear Analysis and Mechanics: Heriot–Watt Symposium, vol. 1, Res. Notes Math., 17, Pitman, 1977, pp. 187–241], or Le Dret [H. Le Dret, Sur les fonctions de matrices convexes et isotropes, CR Acad. Sci. Paris, Série I 310 (1990) 617–620], concerning the convexity of a class of isotropic functions which appear in nonlinear elasticity. Next we prove (Theorem 7.3) a lower semicontinuity result for functionals with the form , with . Here is the usual polar decomposition of , and is Hencky’s logarithmic strain. We close this paper with a compact proof of Dacorogna–Marcellini–Tanteri theorem, based only on classical results about majorization. The mentioned resemblance of this theorem with the Horn–Thompson theorem is thus explained. [Copyright &y& Elsevier]

Abstract: This paper deals with boundary value problems in a bounded domain for the stationary one-dimensional Kawahara equation. For reasonable boundary conditions we prove existence and uniqueness of a strong solution and its asymptotics when the coefficient of a higher order derivative approaches zero. Formulations of well-posed and ill-posed problems for odd-order equations are discussed. [Copyright &y& Elsevier]

In this paper, double dissimilar orthotropic composite materials interfacial crack is studied by constructing new stress functions and employing the method of composite material complex. When the characteristic equations’ discriminants Δ1 > 0 and Δ2 > 0, the theoretical formula of the stress field and the displacement field near the mode I interface crack tip are derived, indicating that there is no oscillation and interembedding between the interfaces of the crack. [ABSTRACT FROM AUTHOR]

The authors consider the simplest quantum mechanics model of solids, the tight binding model, and prove that in the continuum limit, the energy of tight binding model converges to that of the continuum elasticity model obtained using Cauchy-Born rule. The technique in this paper is based mainly on spectral perturbation theory for large matrices. [ABSTRACT FROM AUTHOR]

Weak solution (or generalized solution) for the boundary-value problems of partial differential equations of elasticity of 3D (three-dimensional) quasicrystals is given, in which the matrix expression is used. In terms of Korn inequality and theory of function space, we prove the uniqueness of the weak solution. This gives an extension of existence theorem of solution for classical elasticity to that of quasicrystals, and develops the weak solution theory of elasticity of 2D quasicrystals given by the second author of the paper and his students. [ABSTRACT FROM AUTHOR]

Abstract: This paper is concerned with the existence of a global attractor for a semiflow governed by the weak solutions to a nonlinear one-dimensional thermoviscoelasticity with a non-convex free energy density. The constitutive assumptions for the Helmholtz free energy include the model for the study of martensitic phase transitions in shape memory alloys. To describe physically phase transitions between different configurations of crystal lattices, we work in a framework in which the strain belongs to . New approaches are introduced and more delicate estimates are derived to establish the crucial -estimate of strain in deriving the compactness of the orbit of the semiflow and the existence of an absorbing set. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the regularity, exponential stability of solutions and existence of universal attractor in for a nonlinear one-dimensional compressible Navier–Stokes equations describing a motion of heat-conductive viscous real gas in a bounded domain . Some new ideas and more delicated estimates are introduced to prove these results. [Copyright &y& Elsevier]

In this paper we consider a linearized model for fluid–structure interaction in one space dimension. The domain where the system evolves consists in two parts in which the wave and heat equations evolve, respectively, with transmission conditions at the interface. First of all we develop a careful spectral asymptotic analysis on high frequencies for the underlying semigroup. It is shown that the semigroup governed by the system can be split into a parabolic and a hyperbolic projection. The dissipative mechanism of the system in the domain where the heat equation holds produces a slow decay of the hyperbolic component of solutions. According to this analysis we obtain sharp polynomial decay rates for the whole energy of smooth solutions. Next, we discuss the problem of null-controllability of the system when the control acts on the boundary of the domain where the heat equation holds. The key observability inequality of the dual system with observation on the heat component is derived though a new Ingham-type inequality, which in turn, thanks to our spectral analysis, is a consequence of a known observability inequality of the same system but with observation on the wave component. [Copyright &y& Elsevier]